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The covariation for Banach space valued processes and applications

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Abstract

This article focuses on a recent concept of covariation for processes taking values in a separable Banach space \(B\) and a corresponding quadratic variation. The latter is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace \(\chi \) of the dual of the projective tensor product of \(B\) with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Itô process and the concept of \(\bar{\nu }_0\)-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark–Ocone formula for finite quadratic variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type.

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Acknowledgments

The research was supported by the ANR Project MASTERIE 2010 BLAN-0121-01. The second named author was partially supported by the Post-Doc Research Grant of Unicredit & Universities and his research has been developed in the framework of the center of excellence LABEX MME-DII (ANR-11-LABX-0023-01). The authors are grateful to two anonymous Referees for reading carefully the paper and helping us in improving its quality.

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Authors and Affiliations

  1. Dipartimento di Economia Aziendale, Università G.D’Annunzio di Pescara, Pescara, Italy

    Cristina Di Girolami

  2. Laboratoire Manceau de Mathématiques, Département de Mathématiques, Faculté des Sciences et Techniques, Université du Maine, Avenue Olivier Messiaen, 72085 , Le Mans Cedex 9, France

    Cristina Di Girolami

  3. Département d’Economie, EPEE, Université d’Evry-Val-d’Essonne (TEPP, FR-CNRS 3126), 4 Bd. François Mitterrand, 91025 , Evry cedex, France

    Giorgio Fabbri

  4. ENSTA ParisTech, Unité de Mathématiques appliquées, 828, boulevard des Maréchaux, 91120 , Palaiseau, France

    Francesco Russo

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  1. Cristina Di Girolami
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  2. Giorgio Fabbri
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  3. Francesco Russo
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Correspondence to Francesco Russo.

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Di Girolami, C., Fabbri, G. & Russo, F. The covariation for Banach space valued processes and applications. Metrika 77, 51–104 (2014). https://doi.org/10.1007/s00184-013-0472-6

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